Equilibrium Saddle Points Research Articles

Chaotic dynamics in refraction galactic billiards

We prove the presence of topological chaos at high internal energies for a new class of mechanical refraction billiards coming from Celestial Mechanics. Given an open and bounded domain with smooth boundary, a central mass generates a Keplerian potential in it, while, in , a harmonic oscillator-type potential acts. At the interface, Snell’s law of refraction holds. The chaoticity result is obtained by imposing progressive assumptions on the domain, arriving to geometric conditions which holds generically in . The workflow starts with the existence of a symbolic dynamics and ends with the proof of topological chaos. Intermediate results will be the analytic non-integrability and the presence of multiple heteroclinic connections between different equilibrium saddle points. This work can be considered as the final step of the investigation carried on in De Blasi and Terracini (2022 Nonlinear Anal. 218 112766; 2023 Discrete Contin. Dyn. Syst. 43 1269–318).

Time-inconsistent recursive zero-sum stochastic differential games

In this paper, a kind of time-inconsistent recursive zero-sum stochastic differential game problems are studied by a hierarchical backward sequence of time-consistent subgames. The notion of feedback control-strategy law is adopted to constitute a closed-loop formulation. Instead of the time-inconsistent saddle points, a new concept named equilibrium saddle points is introduced and investigated, which is time-consistent and can be regarded as a local approximate saddle point in a proper sense. Moreover, a couple of equilibrium Hamilton-Jacobi-Bellman-Isaacs equations are obtained to characterize the equilibrium values and construct the equilibrium saddle points.

Dynamic stability, traffic equilibrium and the law of peak-hour expressway congestion

Anthony Downs's (1962) traffic congestion analysis is given a generalized theoretical foundation. It is demonstrated that only two equilibrium saddle points—one stable, the other unstable—can be attained. Once the stable equilibrium is reached, traffic planning and control measures such as drastically increased metering and/or a metropolitan rail transit alternative must be in place so as to maintain equilibrium. If not, the expressway system eventually reaches gridlock. It is also shown that the analyses of Thomson (1977) and Mogridge (1986) parallel Downs's analysis and lead to the same conclusion.

Bifurcations of the trajectories at the saddle level in a Hamiltonian system generated by two coupled Schrodinger equations.

Bifurcations of the complex homoclinic loops of an equilibrium saddle point in a Hamiltonian dynamical system with two degrees of freedom are studied. It arises to pick out the stationary solutions in a system of two coupled nonlinear Schrodinger equations. Their relation to bifurcations of hyperbolic and elliptic periodic orbits at the saddle level is studied for varying structural parameters of the system. Series of complex loops are described whose existence is related to periodic orbits.

Comment on ‘‘Self-nucleation in the sulfuric acid–water system’’

The technique of Doyle [J. Chem. Phys. 35, 795 (1961)] for finding the equilibrium saddle point on the free energy surface of a two-component liquid droplet is criticized. His method is related to modifications of the Laplace equation for the excess pressure inside a droplet and the Kelvin equation for the excess vapor pressure of its constituents. These changes incorrectly account for the variation of surface tension with composition.